Toss arugula and endive in large bowl. (Can be prepared 6 hours ahead. Cover with damp kitchen towel; chill.)
Heat 1 1/2 tablespoons olive oil in large skillet over medium-high heat. Add apples and shallots; sauté 5 minutes. Add chestnuts; sauté 1 minute. Stir in vinegar, scraping up any browned bits. Remove from heat; stir in walnut oil and remaining 1 1/2 tablespoons olive oil. Season with salt and pepper. Pour chestnut mixture over arugula mixture; toss. Divide salad among 8 plates.